\(\bullet \)  If \(\mu =\int \mu _x\, d\mu \)  is the ergodic decomposition of the measure \(\mu \) , it is known that if \(\vert \!|\!| f|\!|\!\vert _{s,\mu }=0\) , then \(\vert \!|\!| f|\!|\!\vert _{s,\mu _x}=0\)  for \(\mu \) -almost every \(x\in X\)  [1]}. As a consequence, if a collection of sequences is good for seminorm estimates for every ergodic system, then it is also good for seminorm estimates for arbitrary systems.
